Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice

Abstract

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12)^2$ ``Super-Kagome'' lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an ``all-or-nothing'' phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other.

Figures (6)

• Figure 1: The two Archimedean tilings primarily investigated in this article.
• Figure 2: Fundamental domain of the Kagome lattice with edge weights. In the monomeric case, all edge weights around downwards pointing triangles are $\gamma_2 = \gamma_4 = \gamma_6 =: \alpha$ and all edge weights on upwards pointing triangles are $\gamma_1 = \gamma_3 = \gamma_5 =: \beta$, where $2 \alpha + 2 \beta = \mu$.
• Figure 3: Spectrum of the monomeric $(3^2.6^2)$ Kagome lattice with vertex weight $\mu > 0$ as a function of the parameter $\alpha \in (0, \frac{\mu}{2})$, describing the edge weights on edges adjacent to downwards pointing triangles.
• Figure 4: Fundamental domain of the $(3.12^2)$ tiling with edge weights. In the monomeric case, all edge weights around triangles triangles are $\gamma_1 = \dots = \gamma_6 =: \alpha$ and the remaining weights are $\gamma_7 = \gamma_8 = \gamma_9 =: \beta$.
• Figure 5: Spectrum of the monomeric $(3.12^2)$ "Super-Kagome" lattice with vertex weight $\mu > 0$ as a function of the parameter $\alpha \in (0, \frac{\mu}{2})$, describing the edge weights on edges adjacent to triangles.

Theorems & Definitions (19)

• Lemma 1
• Remark 2
• Definition 3
• Proposition 4: See TaeuferPeyerimhoff and references therein
• Proposition 5
• Theorem 6
• Lemma 7
• proof
• proof : Proof of Theorem \ref{['thm:FlatBandsKagome']}
• Theorem 8: Band gaps in the perturbed Kagome lattice